It takes 9 hours for Isabella to rake leaves by herself, but her brother Matthew can work three times as fast. If they work together, how long it will take them to rake leaves?

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lemjay | High School Teacher | (Level 2) Senior Educator

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For 1 complete job, it takes 9 hour for Isabella to finish it alone.
So, her rate is:

`Rate=(Work d o n e)/(time)`

`R_I=1/9`

Since Matthew works three times as fast as compared to Isabella, then his rate is:

`R_M=3*R_I=3*1/9`

`R_M=1/3`

If they work together to do 1 complete job, their total rate is:

`R_T=R_I+R_M`

Since total rate is also equal to work done ( which is assumed as 1 complete job) divide by total time `(R_T=1/T)` ,then the equation is:

`1/T =1/9+1/3`

`1/T=1/9+3/9`

`1/T=4/9`

To simplify the equation, multiply both sides by the LCD of the two fractions which is 9T` `.

`9T*1/T=4/9*9T`

`9=4T`

Then, divide both sides by 4 to isolate T.

`9/4=(4T)/4`

`9/4=T`

`2.25=T`

Hence, working together, it will take them 2.25 hours to rake the leaves.

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